Gauge theory - Imported Knowlpedia Content

Definition

A classical geometric gauge theory on a manifold $M$ specifies a principal bundle $P\to M$ , a configuration space containing principal connections on $P$ and possibly additional matter fields, and an action of the gauge group $\mathcal G(P)$ . Its equations, action functional, and observables are required to be invariant under this action. Configurations related by a gauge transformation represent the same physical or geometric state, so solutions are studied through their gauge-equivalence classes. This definition describes the classical bundle-theoretic framework; a particular gauge theory also fixes the structure group, field content, functional, boundary conditions, and regularity class.

Geometric structure

A connection is the gauge potential, while its curvature is the corresponding field strength. Matter fields are commonly sections of bundles associated to $P$ , and a connection on $P$ induces covariant derivatives on those bundles. Gauge transformations act simultaneously on the connection and matter fields, preserving the theory's geometric constructions.

The quotient of the solution set by $\mathcal G(P)$ is a moduli space. It need not be a smooth manifold: reducible configurations have nontrivial stabilizers, and nonlinear equations can produce singularities.

Yang–Mills example

Given a Riemannian metric on $M$ and an invariant inner product on the Lie algebra of a compact structure group, the Yang–Mills functional is

\operatorname{YM}(A)=\frac12\int_M \langle F_A,F_A\rangle\,\mathrm{vol}_M.

Its critical points are Yang–Mills connections. The functional is gauge invariant because curvature transforms by the adjoint action and the inner product is invariant. Coupling sections of associated bundles to $A$ produces standard gauge–matter systems.

Conventions and scope

The word “gauge theory” covers more than Yang–Mills theory. Topological gauge theories, theories with higher-form gauge fields, and discrete gauge theories can require different geometric models.

References

M. J. D. Hamilton, Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics, Springer, 2017. DOI record. Relevant: principal bundles, connections, gauge transformations, and Yang–Mills theory.
J. C. Baez and J. P. Muniain, Gauge Fields, Knots and Gravity, World Scientific, 1994. DOI record. Relevant: chapter 2, the bundle-theoretic formulation of gauge fields.